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Sum up to 16 terms of the series (1^(3))...

Sum up to 16 terms of the series `(1^(3))/(1) + (1^(3) + 2^(3))/(1 + 3) + (1^(3) + 2^(3) + 3^(3))/(1 + 3 + 5) + ..` is

A

450

B

456

C

446

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`t_(n) = (1^(3) + 2^(3) + 3^(3) + .... + n^(3))/(1 +3 + 5 + ...(2n -1)) = ({(n(n + 1))/(2)}^(2))/((n)/(2) {2 + 2 (n -1)}) = ((n^(2) (n + 1)^(2))/(4))/(n^(2)) = ((n+1)^(2))/(4) ==(n^(2))/(4) + (n)/(2) + (1)/(4)`
`:. S_(n) = Sigmat_(n) = (1)/(4) Sigman^(2) + (1)/(2) Sigman + (1)/(4) Sigma 1 = (1)/(4) .(n(n + 1)(2n + 1))/(6) + (1)/(2). (n(n + 1))/(2) + (1)/(4). n`
`:. S_(16) = (16.17.33)/(24) + (16.17)/(4) + (16)/(4) = 446`
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